What do the letters after a UK clinician's name mean? MRCPsych, GMC number, ANP?

On a UK clinician profile, those letters are not interchangeable. A GMC number identifies a doctor's registration with the General Medical Council. The public medical register shows GMC number and registration status; that registration is the licence to practise medicine, not a psychiatry diploma on its own. References: https://www.gmc-uk.org/registration-and-licensing/the-medical-register https://www.rcpsych.ac.uk/ https://www.nmc.org.uk/registration/search-the-register/ https://www.nice.org.uk/guidance/ng87/chapter/recommendations